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1 in 6 wins As a special promotion for its 20 -ounce bottles of soda, a soft drink company printed a message on the inside of each cap. Some of the caps said, "Please try again," while others said, "You're a winner!' The company advertised the promotion with the slogan "l in 6 wins a prize." Suppose the company is telling the truth and that every 20 -ounce bottle of soda it fills has a 1 -in-6 chance of being a winner. Seven friends each buy one 20 -ounce bottle of the soda at a local convenience store. Let \(X=\) the number who win a prize. (a) Explain why \(\mathrm{X}\) is a binomial random variable. (b) Find the mean and standard deviation of \(\mathrm{X}\) . Interpret each value in context. (c) The store clerk is surprised when three of the friends win a prize. Is this group of friends just lucky, or is the company's 1 -in-6 claim inaccurate? Compute \(P(\mathrm{X} \geq 3)\) and use the result to justify your answer.

Step by step solution

01

Understanding the Problem

We need to determine the nature of the random variable, the mean and standard deviation, and assess the likelihood of an observed outcome. We'll approach this in steps suitable for a binomial probability scenario.

02

Identify Characteristics of X

The random variable \(X\) is the number of friends who win a prize out of seven. The conditions for a binomial distribution are met if: each friend encounters a binary outcome (win or lose), the probability of winning is constant at \(\frac{1}{6}\), and events (each friend's win) are independent.

03

Confirm X as a Binomial Variable

\(X\) follows a binomial distribution because there are a fixed number of trials (7 friends), each with two possible outcomes (win or lose), a constant probability of success \(p = \frac{1}{6}\), and independent trials. Hence, \(X \sim \text{Binomial}(7, \frac{1}{6})\).

04

Calculate Mean and Standard Deviation

The mean \(\mu\) of a binomial distribution is \(\mu = np\) where \(n\) is the number of trials, and \(p\) is the probability of success. Therefore, \(\mu = 7 \times \frac{1}{6} = \frac{7}{6}\). The standard deviation \(\sigma\) is given by \(\sigma = \sqrt{np(1-p)}\), so \(\sigma = \sqrt{7 \times \frac{1}{6} \times \frac{5}{6}} = \sqrt{\frac{35}{36}}\approx 0.983\). The mean represents the expected number of winners, while the standard deviation indicates the typical deviation from the mean.

05

Find Probability of 3 or More Winners

We need to compute \(P(X \geq 3)\). Using the binomial probability formula or calculator, we find \(P(X \geq 3) = 1 - P(X < 3)\). Calculate \(P(X = 0), P(X = 1), P(X = 2)\), then \(P(X \geq 3) = 1 - (P(X = 0) + P(X = 1) + P(X = 2))\). This evaluates to approximately 0.086.

06

Interpret the Probability Result

Since \(P(X \geq 3)\approx 0.086\), which is unlikely (less than 0.1), it suggests that having 3 winners is fortunate though not extremely rare. The occurrence is possible, but a higher probability variance could imply unusual luck or inaccuracies in the company's claim.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability

Probability helps us understand how likely an event is to happen. In the context of this exercise, probability comes into play as we calculate the chances of winning a prize from purchasing a soda bottle. If each bottle has a 1-in-6 chance of winning, this means:

• The probability of winning a prize with one bottle is \( \frac{1}{6} \).
• Conversely, not winning a prize has a probability of \( \frac{5}{6} \).

When calculating probabilities in a binomial distribution (like our case), we are determining how likely it is to get a certain number of 'successes' (winnings) across a series of trials (purchases). This exercise asks you to find the probability of 3 or more wins among 7 friends, utilizing the binomial distribution. Understanding these basic probability principles helps you interpret different scenarios and outcomes effectively.

Random Variable

In probability, a random variable represents a measurable outcome resulting from a random process. In this exercise, we use a random variable, say \(X\), to express the number of times an outcome occurs.
Here, \(X\) is the number of friends who win a prize from buying soda bottles.

Random variables can be classified into types, namely:

• Discrete Random Variables: These take on a countable number of possible values, like the number of winning friends (0 to 7) in our example.
• Continuous Random Variables: These assume any value within a range, typically involving measurements with infinite possibilities, not applicable in our discrete situation.

Understanding that each friend's bottle purchase and outcome is independent makes \(X\) a perfect example of how random variables help model real-world scenarios, leading us toward meaningful conclusions about the distribution of outcomes.

Mean and Standard Deviation

The mean and standard deviation are measures used to describe the characteristics of a probability distribution.
For a binomial distribution like the one with 7 trials and a success probability of \( \frac{1}{6} \), they help summarize anticipated outcomes:

• Mean (\( \mu \)): This is the expected average number of successes. It is computed as \( np = 7 \times \frac{1}{6} = \frac{7}{6} \). This value means you can expect about 1.17 winning friends on average over many repeat events of this same experimental setup.
• Standard Deviation (\( \sigma \)): This measures the distribution's spread. It indicates how much the number of winners typically differs from the mean, calculated as \( \sqrt{np(1-p)} = \sqrt{7 \times \frac{1}{6} \times \frac{5}{6}} \approx 0.983 \).

The smaller the standard deviation, the more closely the results are expected to cluster around the mean value. However, this scenario shows a moderate deviation, signifying possible variance in outcomes from expectation.

Binomial Random Variable

A binomial random variable emerges in situations with fixed numbers of independent trials, each having two outcomes: success or failure. This tightly ties into our present challenge, where \(X\) is a binomial random variable. Here's why:

• Fixed Number of Trials: We observe a specific 7 trials, representing each friend's purchase of one bottle.
• Binary Outcomes: Each bottle can either be a winner (success) or not (failure).
• Constant Success Probability: Every trial carries a \( \frac{1}{6} \) chance of a successful outcome.
• Independence: Each friend's result does not influence another's due to the fixed probability and separate trials.

Binomial variables are key to answering questions about the likelihood of observing a particular number of successes and become crucial when computing probabilities like \( P(X \geq 3)\), where interpretation of results can be utilized to assess if outcomes like the soda cap slogan's reliability stand true.